Functions?

Identify the open intervals on which the function is increasing or decreasing. (Enter your answers using interval notation.)
h(x) = 7sqrt(x) e^(−x)
Increasing: ???
decreasing:???

1 Answer

The function h(x)=7sqrtxe^(-x) is increasing in open interval (1/2,oo) and decreasing in open interval (0,1/2)

Explanation:

We have h(x)=7sqrtxe^(-x). Observe that it is not defined for x<0.

Now as (dh)/(dx)=7[e^(-x)/(2sqrtx)-sqrtxe^(-x)]=(7e^(-x))/(2sqrtx)(1-2x)

(dh)/(dx) >0 for x<1/2 and (dh)/(dx) <0 for x>1/2

and hence the function h(x)=7sqrtxe^(-x)

is increasing in open interval (0,1/2)

and decreasing in open interval (1/2,oo)

graph{7sqrtxe^(-x) [-3.063, 6.937, -1.06, 3.94]}